The Number Theory Method: Divisibility, Congruences, Primes, and Cryptography | Step-by-Step Exercises with Full Solutions

Learn number theory as a method for reasoning with the integersAre the prime numbers infinite in count? Euclid answered in four lines, nearly 2300 years ago, and his proof remains one of the most perfect in mathematics: assume the list of primes ends at some prime — multiply them all together, add 1, and you obtain a number divisible by none of the primes on your list. That number is therefore either itself prime, or divisible by a prime missing from your list. Either way, your list was incomplete. This piercing elegance, at the heart of Chapter 2, sets the tone for the entire book. From the Euclidean algorithm to the RSA keys that secure your phone every day, this book takes you through elementary number theory — by hand, without software, through rigorous reasoning alone.Across 8 progressive chapters, you discover divisibility, gcd and Bezout's identity with the Extended Euclidean Algorithm, the prime numbers and the Fundamental Theorem of Arithmetic with Euclid's proof of their infinitude, modular arithmetic and linear congruences with the modular inverse, the Chinese Remainder Theorem and the complete solution of Sun Tzu's 1500-year-old puzzle, the theorems of Fermat, Euler, and Wilson with the totient function phi, primitive roots and the discrete logarithm as a bridge to Diffie–Hellman, quadratic residues and Gauss's law of quadratic reciprocity (the "golden theorem"), and finally RSA cryptography rebuilt by hand from the choice of two primes to the full decryption, with Pell's equation and sums of two squares. Three recipes in 4 or 5 steps structure the method.What's inside8 progressive chapters, from divisibility to modern cryptography26 worked examples in full detail to anchor every technique72 exercises with complete, step-by-step solutions3 structuring recipes: Extended Euclidean Algorithm, Chinese Remainder Theorem, RSA key generationEuclid's proof of the infinitude of primes — four lines, an elegance intact for 2300 yearsThe Chinese Remainder Theorem applied step by step to Sun Tzu's puzzleGauss's law of quadratic reciprocity illustrated through concrete computationsThe RSA cryptosystem rebuilt by hand, from choosing two primes to full decryptionWhy this bookA method, not a treatise: you learn to reason with the integers, not to memorize theoremsThe three structuring recipes: the tools every mathematician and cryptographer actually usesFour-step pedagogy: concept, example, exercise, solutionHand computation exclusively: no software required, understanding guaranteedSuited for self-study, undergraduate and graduate programs in mathematics, computer science, or engineeringWhether you are pursuing a degree in mathematics, computer science, or cryptography, a professional who wants to understand the arithmetic foundations of digital security, or a self-learner captivated by the primes — this book is your method. A few chapters are enough to transform the way you think about the integers and modern cryptography. Read more